From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 29 Oct 2018 05:54:43 +0000 (+1100)
Subject: vectors ref
X-Git-Tag: yr11~8
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/baa61484fba8c0936cc28ba629475821dfd34936

vectors ref
---

diff --git a/spec/vectors.md b/spec/vectors.md
index 8fa7208..dcdf2c5 100644
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -1,18 +1,19 @@
 ---
-header-includes:
-  - \documentclass{standalone}
-  - \usepackage{cleveref}
-  - \usepackage{harpoon}
-  - \usepackage{accent}
-  - \usepackage{amsmath}
-...
+geometry: margin=2cm
+<!-- columns: 2 -->
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+
+---
 
 # Vectors
 
 - **vector:** a directed line segment  
 - arrow indicates direction
 - length indicates magnitude
-- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
+- notated as $\vec{a}, \widetilde{A}$
 - column notation: $\begin{bmatrix}
        x \\ y
      \end{bmatrix}$
@@ -87,6 +88,19 @@ Produces a real number, not a vector.
 
 $$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
 
+## Scalar product properties
+
+1. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$
+2. $\boldsymbol{a \cdot 0}=0$
+3. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
+
+For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:  
+$\boldsymbol{a \cdot b}=\{
+                \begin{array}{ll}
+                  |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction} \\
+                  -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions} \\
+                \end{array}$
+
 ## Geometric scalar products
 
 $$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$
diff --git a/spec/vectors.pdf b/spec/vectors.pdf
new file mode 100644
index 0000000..b60fffc
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